Finite Group Databasebeta

← #29707312

Subgroups — #29707312

Order 15616.

1498 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16839760C212yesmaximal
3#21448306C211yesmaximal, characteristic
4#21446961C211yesmaximal, characteristic, complemented
5#21446959C212yesmaximal, complemented
6#21446957C211yesmaximal, characteristic, complemented
7#21446955C211yesmaximal, characteristic, complemented
8#21446953C212yesmaximal, complemented
9#21445537C212yesmaximal, complemented
10#21445511C211yesmaximal, characteristic, complemented
11C61×#2023C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×Q8×D61—21no—
14C22×#11758189C2212yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519776—21no—
17C2×#14519776C2212yes—
18C2×#14519775—21no—
19C4⋊Q8×D61—28no—
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16839760—28no—
25C2×#14519702C2212yes—
26C2×#14519700C2212yes—
27C2×#14519698C2212yes—
28#16839668C2211yescharacteristic
29#16839667C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839638—21no—
32#16839638—21no—
33C22⋊C4×Dic61—21no—
34C22⋊C4×Dic61—21no—
35#16839525C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C61×#390C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C2×#11758224—24no—
46C2×#11758224—24no—
47C2×#11758224—22no—
48C2×#11758224Q814yescomplemented
49C22×Dic122—21no—
50C22×Dic122—22no—
51C22×Dic122C2311yescharacteristic
52C2×Q8×D61—28no—
53C2×Q8×D61—44no—
54C2×Q8×D61—44no—
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C22×C4×D61—22no—
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C2×#11758189—24no—
62C2×#11758189—24no—
63C2×#11758189—24no—
64C22×#92418C2312yes—
65#14519777—24no—
66#14519776—28no—
67#14519776—24no—
68#14519776—24no—
69#14519775—28no—
70#14519735—24no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C4⋊C4×D61—28no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61D418yes—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61—24no—
87#14519702—24no—
88#14519700—24no—
89#14519698—24no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C22×Q8×C61—21no—
95C22×D4×C61C2311yescharacteristic
96C22×C8×C61C2312yes—
97C4⋊Q8×C61—24no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#102×C61—24no—
105C24×C61—21no—
106C23×D61C2×D411yescharacteristic
107C2×Q8×C61—24no—
108C2×Q8×C61—42no—
109C2×Q8×C61—42no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61C2×Q812yescomplemented
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—42no—
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—24no—
125C22×Dic61—21no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131Q8×D61—432no—
132#11758224—216no—
133#11758224—48no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—28no—
141C2×C4×D61—44no—
142C2×C4×D61—44no—
143C2×C4×D61C2×D414yes—
144C2×C4×D61C2×Q812yes—
145C2×C4×D61#4114yes—
146C2×C4×D61#4112yes—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—24no—
150C2×Dic122—28no—
151C2×Dic122—21no—
152C2×Dic122—44no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×Dic122—44no—
156C2×Dic122C2×Q812yescomplemented
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C2×D414yes—
164C22⋊C4×C61—42no—
165C22⋊C4×C61—42no—
166C42×C61—22no—
167C22⋊Dic61—42no—
168C22⋊Dic61—42no—
169#11758198—42no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61C2×D414yes—
174#11758196—28no—
175#11758196—28no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178C4×Dic61—22no—
179C4×Dic61—42no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183#11758189—48no—
184C23×C61—22no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61C22×D411yescharacteristic
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61#12012yes—
197C22×D61C22⋊Q812yes—
198C22×D61#12412yes—
199Q8×C61—48no—
200D4×C61—24no—
201D4×C61—42no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—24no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61AGL(1,ℤ/8)12yes—
213C2×C4×C61#13712yes—
214C2×C4×C61C22×D412yes—
215C2×C4×C61C22×Q811yescharacteristic
216C2×C4×C61C2×#4112yes—
217C2×C4×C61C2×#4111yescharacteristic
218C22⋊D61—48no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—24no—
224C2×Dic61—22no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—48no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61#12012yes—
234C2×Dic61C22⋊Q812yes—
235C2×Dic61#12412yes—
236C2×Dic61AGL(1,ℤ/8)12yes—
237C2×Dic61#13712yes—
238C4×D61—28no—
239C4×D61—28no—
240C4×D61—416no—
241C4×D61C22⋊Q818yes—
242C4×D61#12418yes—
243Dic122—24no—
244Dic122—416no—
245Dic122—48no—
246Dic122—42no—
247C8×C61—42no—
248#92418—42no—
249#23452—611nomaximal, Sylow
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C2×#12011yescharacteristic
258C22×C61C2×C22⋊Q811yescharacteristic
259C22×C61C2×#12411yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61C2×#13711yescharacteristic
263C22×C61C2×#13711yescharacteristic
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61#39314yes—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—44no—
274C4×C61—81no—
275C4×C61C2×C22⋊Q812yes—
276C4×C61C2×#12412yes—
277Dic61—22no—
278Dic61—22no—
279Dic61—42no—
280Dic61—44no—
281Dic61—81no—
282Dic61#39314yes—
283C22×C4⋊Q8—611no—
284C2×#417—612no—
285#2297—611no—
286#2024—612no—
287#2024—611no—
288#2024—611no—
289#2023—612no—
290#2023—611no—
291#2023D6111yescharacteristic, complemented
292#1736—612no—
293#1728—611no—
294C2×C61—82no—
295C2×C61C2×#39311yescharacteristic
296C2×C61#300311yescharacteristic
297C2×C61#300411yescharacteristic
298C2×C61#307911yescharacteristic
299C2×C61#307911yescharacteristic
300C2×C61#314111yescharacteristic
301C2×C61#314111yescharacteristic
302D61—24no—
303C22×#41—611no—
304C23×Q8—1221no—
305C22×M16—612no—
306C2×C4⋊Q8—612no—
307C2×C4⋊Q8—611no—
308C2×C4⋊Q8—1221no—
309C2×C4⋊Q8—1221no—
310C2×C4⋊Q8—1228no—
311C2×C4⋊Q8C2×D6111yescharacteristic
312C2×#117—611no—
313C22×C4⋊C4—612no—
314C22×C4⋊C4—1221no—
315C22×C42—611no—
316#417—1228no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#102—612no—
320C2×#102C2×D6112yes—
321#391—611no—
322#391—611no—
323#390—611no—
324#390C2×D6111yescharacteristic
325#386—1221no—
326#386—1221no—
327C4×C22⋊C4—1221no—
328C4×C22⋊C4—1221no—
329#340—612no—
330#340—612no—
331#340—612no—
332#340C2×D6112yes—
333#336—612no—
334#336—611no—
335#336C2×D6111yescharacteristic
336C61#2345211yescharacteristic, complemented, Sylow
337C2×#41—614no—
338C2×#41—1222no—
339C2×#41—1224no—
340C2×#41—1224no—
341C22×Q8—21no—
342C22×Q8—611no—
343C22×Q8—1222no—
344C22×Q8—1228no—
345C22×Q8—1221no—
346C22×Q8—2444no—
347C22×Q8—2444no—
348C22×D4—1221no—
349C22×D4C22×D6111yescharacteristic
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—1222no—
354C23×C4—1221no—
355C2×M16—1224no—
356C2×M16—1224no—
357C2×M16—1224no—
358C22×C8—612no—
359C22×C8C22×D6112yes—
360C4⋊Q8—24no—
361C4⋊Q8—1224no—
362C4⋊Q8—1224no—
363C4⋊Q8—1228no—
364C4⋊Q8—1228no—
365C4⋊Q8—1224no—
366#117—1224no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—618no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1228no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6112yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—1224no—
387C2×C42—1221no—
388C2×C42C22×D6111yescharacteristic
389#103—1224no—
390#103—1224no—
391#102—24no—
392#102—1224no—
393#95—1221no—
394#95—1221no—
395#95—1221no—
396#95—1221no—
397C24—21no—
398C24—611no—
399#41—12216no—
400#41—2448no—
401C2×Q8—24no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—612no—
405C2×Q8—1224no—
406C2×Q8—1228no—
407C2×Q8—1221no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—2442no—
411C2×Q8—2444no—
412C2×Q8—2444no—
413C2×Q8—24432no—
414C2×Q8—2442no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—1224no—
419C2×D4—2442no—
420C2×D4—2442no—
421C2×D4Q8×D6112yescomplemented
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—611no—
425C22×C4—614no—
426C22×C4—612no—
427C22×C4—614no—
428C22×C4—612no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1228no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1221no—
440C22×C4—1222no—
441C22×C4—1221no—
442C22×C4—2444no—
443C22×C4—2444no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4D4×D6111yescharacteristic
447C22×C4D4×D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449M16—2448no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—614no—
460C4⋊C4—1228no—
461C4⋊C4—1228no—
462C4⋊C4—1224no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4D4×D6114yes—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—2442no—
469C22⋊C4—2442no—
470C22⋊C4—2442no—
471C42—22no—
472C42—1222no—
473C42—1222no—
474C42—1222no—
475C42—2442no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—612no—
484C23—612no—
485C23—612no—
486C23—1221no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23C2×D4×D6111yescharacteristic
491Q8—48no—
492Q8—1224no—
493Q8—2448no—
494Q8—24416no—
495Q8—2442no—
496D4—24no—
497D4—42no—
498D4—2448no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—618no—
513C2×C4—618no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—1228no—
517C2×C4—1228no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1224no—
523C2×C4—1222no—
524C2×C4—24416no—
525C2×C4—2448no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4#1451980112yes—
534C2×C4#1451980612yes—
535C2×C4C2×D4×D6112yes—
536C2×C4C2×Q8×D6111yescharacteristic
537C2×C4#41×D6112yes—
538C2×C4#41×D6111yescharacteristic
539C8—42no—
540C8—2442no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—614no—
549C22—1222no—
550C22—1222no—
551C22—1222no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22#120×D6111yescharacteristic
556C22C22⋊Q8×D6111yescharacteristic
557C22#124×D6111yescharacteristic
558C22AGL(1,ℤ/8)×D6111yescharacteristic
559C22C2×#1451980111yescharacteristic
560C22#137×D6111yescharacteristic
561C22C2×#1451980611yescharacteristic
562C4—22no—
563C4—22no—
564C4—44no—
565C4—81no—
566C4—614no—
567C4—1222no—
568C4—1222no—
569C4—4881no—
570C4—2442no—
571C4—2444no—
572C4C22⋊Q8×D6112yes—
573C4#124×D6112yes—
574C2—82no—
575C2—1224no—
576C2#393×D6111yescharacteristic
577C2#2145504311yescharacteristic
578C2#2145505011yescharacteristic
579C2#2145562711yescharacteristic
580C2#2145563111yescharacteristic
581C2#2145620911yescharacteristic
582C2#2145621111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2023C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#2023D61
hypocenterC61#23452
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C4⋊Q8×D61C2semidirect
4. #21446961C2semidirect
5. #21446959C2semidirect
6. #21446957C2semidirect
7. #21446955C2semidirect
8. #21446953C2semidirect
9. #21445537C2semidirect
10. #21445511C2semidirect
11. C61×#2023C2semidirect
48. C2×#11758224Q8semidirect
113. C2×D4×C61C2×Q8semidirect
156. C2×Dic122C2×Q8semidirect
291. #2023D61semidirect
336. C61#23452semidirect
421. C2×D4Q8×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4⋊Q8×D612
C2×#168397602
#214483062
#214469612
#214469592
#214469572
#214469552
#214469532
#214455372
#214455112
C61×#20232
#2345261